Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1905 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #386 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,17,14,19,12,18)(10,20,23,30,24,33,22,15)(11,25,27,16,28,31,26,32)(21,29,36,38,37,39,35,34);(8,10)(9,15,19,30)(11,21)(12,22)(13,23)(14,24)(16,29,32,39)(17,20,18,33)(25,34,31,38)(26,35)(27,36)(28,37);(8,11,14,28)(9,16,19,32)(10,21,24,37)(12,27,13,26)(15,29,30,39)(17,25,18,31)(20,34,33,38)(22,36,23,35)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5140);